In a tappet of the aforesaid type known from German Utility Model G 93 17 325.3, the two bush-like sections of the skirt are joined to each other by welding, soldering or gluing, and the tappet is heat-treated as a whole to obtain higher hardness values. A drawback of this prior art is that the joining of the two skirt sections by the joining methods mentioned can result in an obstruction of the oil supply channel with the known negative consequences. A further drawback is that thin-walled regions of the tappet are hardened to the core during the heat treatment and this leads to their embrittlement and a tendency to crack. Methods used in the past to avoid these problems consisted in either shielding the thin-walled parts during the heat treatment or re-annealing them after the heat treatment. This procedure, however, involves more time and work and is therefore more expensive.